The incident happened at the Co-op in Bomere Heath at about 4.14am, and left the front windows and door damaged.
It is believed that the suspects attached chains to the ATM before attempting to remove it using a vehicle which failed.
A police officer was guarding the premises where food and drink can been scattered on the floor inside the store.
The front window was left with a large crack and the door and its frame damaged as a result of the attack.
Police said a white van was then seen leaving the area at speed, and asked for anybody who had seen it to come forward.
Co-op spokesman Andrew Torr said: “There was an incident at our Bomere Heath store in the early hours of this morning where there was an unsuccessful attack on the community’s ATM.
"We appeal for anyone who may or seen anything to come forward to the police who are investigating. It is too early to confirm when the store will reopen to serve the community while repairs are made to make the store safe and secure and, we await the outcome of structural assessment.”
He said any CCTV footage will be handed to West Mercia Police as part of an investigation.
The chain said it is investing millions of pounds in new security technology to protect its staff and communities. This includes working with Shropshire-based security goods company Smartwater to develop its new fog cannon which is aimed to protecting tobacco and alcohol stock and Smartwater gels in cash machines which leave marks on thieves.
Alex Gale, West Mercia Police spokesman, said: "The incident happened at around 4.15am when several offenders tried to gain access to the store. The offenders did not get away with any cash but did cause a significant amount of damage to the shop.
"A white van was seen leaving the scene at speed and we would like to hear from anyone who may have seen or captured any of the incident on the dash cam or mobile phone."
Witnesses with information can phone 101 quoting incident 91s of April 18 2019.
Alternatively, contact Crimestoppers anonymously on 0800 555111 or at crimestoppers-uk.org